Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems

Abstract

This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say \(q(\tau ,A)\), on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of \(q(\tau ,w)\) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.